Global dynamical structures from infinitesimal data

TL;DR

The paper addresses the challenge of uncovering global dynamical structure from high-dimensional trajectory data without committing to a full global model. It proposes the ILG framework implemented as VERT, which leverages local linearization and a fiberwise distance estimator $f(z) := \vert H(z) v(z) \vert$ to locate attracting sets tied to the system's asymptotic behavior. Key contributions include formal accuracy guarantees, applicability to both continuous and hybrid dynamics, and demonstrations on synthetic multiscale models as well as human running data to identify candidate control modules. This model-agnostic approach enables rigorous inference of underlying dynamical structure in systems where global dynamics are difficult or impossible to learn, with broad implications for biology and engineering.

Abstract

Scientists and engineers alike target modeling of complex, high dimensional, and nonlinear dynamical systems as a central goal. Machine learning breakthroughs alongside mounting computation and data advance the efficacy of learning from trajectory measurements. However scientifically interpreting data-driven models, e.g., localizing attracting sets and their basins, remains elusive. Such limitations particularly afflict identification of system-level regulatory mechanisms characteristic of living systems, e.g., stabilizing control for whole-body locomotion, where discontinuous, transient, and multiscale phenomena are common and prior models are rare. As a next step towards theory-grounded discovery of behavioral mechanisms in biology and beyond, we introduce VERT, a framework for discovering attracting sets from trajectories without recourse to any global model. Our infinitesimal-local-global (ILG) pipeline estimates the proximity of any sampled state to an attracting set, if one exists, with formal accuracy guarantees. We demonstrate our approach on phenomenological and physical oscillators with hierarchical and impulsive dynamics, finding sensitivity to both global and intermediate attractors composed in sequence and parallel. Application of VERT to human running kinematics data reveals insight into control modules that stabilize task-level dynamics, supporting a longstanding neuromechanical control hypothesis. The VERT framework promotes rigorous inference of underlying dynamical structure even for systems where learning a global dynamics model is impractical or impossible.

Figures (5)

• Figure 1: Overview of the VERT computational framework. Panels (A-B) depict the relation to motivating hierarchical control hypotheses for complex, multi-scale systems, viewed through an illustrative application setting of the postural dynamics of human running. Panels (D-F) outline the steps of the VERT algorithm. A. Stable task-level system behaviors such as running emerge from the coupled, highly nonlinear, closed loop dynamics of many body segments and their interaction with the environment. B. We hypothesize that for many tasks, the global task-level behavior may be decomposed into hierarchically arranged control modules, whose coordinated interactions characterize the dynamics of appendages and body segments. The control module state spaces are embedded as submanifolds (postures) that carry the constituent dynamics (templates) of the full, coordinated system. C. We can use VERT to learn a spatiotemporal filter on the trajectory data that isolates locally stable (template) behavior from transient behaviors reflecting disturbances and stabilizing control (anchoring). D. VERT estimates the distance from each sample to the attractor guiding its asymptotic behavior ($f(z)$, Def. \ref{['def:sensingfunction']}). The estimation procedure for a point $z^*$ begins with identification of its metric nearest neighbors in state space. E. A Cartesian basis for the tangent space at $z^*$ of the manifold containing the trajectories is estimated. The local subsample of the vector field (temporal differences, gray arrows) is projected onto the $n$-dimensional local vielbein basis (black arrows). F. Using this representation, we can estimate the component of the vector field on the vertical subspace, the local trivialization of the global, typically nonlinear vertical subbundle of the attractor. We can use this estimate to approximate the distance from $z^*$ to its base point on the unknown attractor $\mathcal{A}$. Steps D-F are repeated for an appropriate number of samples. Filtering the trajectory data for sublevel sets of $f(z)$ reveals the posture carrying the template dynamics (blue) and hierarchical stabilization (anchoring) intervals of what subsequent hypothesis-informed analysis can posit as the stance leg and upper body posture during stance (See Fig. \ref{['fig:human']}).
• Figure 2: Application of VERT to the A-HM model illustrates the mechanisms laid out in Fig. \ref{['fig:hopfresults']} that underlie the hypotheses in Table \ref{['table:hypotheses']}: the distance estimator is sensitive to the locally dominant dynamics at the sampled point. A. Hierarchical composition of the A-HM (Eqns. \ref{['eq:hopf']} and \ref{['eq:anchor']}). The cross product system of two limit cycles produces a limit torus, whose product with a higher dimensional stable spiral generates qualitatively distinct mechanisms for anchoring the global attractor. B. Visualization of VERT estimator ($f$) in the error coordinate plane. Observe the initial alignment of the $f$-level sets with the error coordinate of the stronger subsystem. Upon reaching an intermediate attractor (at vertical and horizontal axes), $f$ continues to decay more slowly to its global minimum. C. Sensitivity plots of $f$ for different relative subsystem gains. When $\gamma_\rho = \gamma_\delta$, $f$ varies linearly with both error coordinates. When one gain is larger, $f$ shows heightened sensitivity to the stronger system.
• Figure 3: Parameter sweep experiments illustrate sensitivity of the VERT estimator to attractor hierarchies. See Table 1 for hypothesis definitions A. When $\gamma_\theta=0$, the limit cycle $\mathcal{S}$ does not exist, and the outputs of VERT accordingly support rejection of $H_0$. $H_{1-3}$ are supported, as the residual decreases with both $(\gamma_\rho, \gamma_\delta)$ for $\mathcal{T}$, $\gamma_\delta$ for $\mathcal{P}$, and $\gamma_\rho$ for $\mathcal{C}$. B. When $\gamma_\theta=1$, $\mathcal{S}$ exists and is detected. $H_{0-3}$ are supported, with similar behavior to the $\gamma_\theta$ for the intermediate submanifolds.
• Figure 4: VERT predicts locally stable regions across continuous and hybrid behaviors in the van der Pol system. A. Hierarchical and sequential attractors of Eq. \ref{['eq:vdp']} across the nonlinear damping parameter $\mu$. The grey plane $(x_1, x_2)$ contains the van der Pol vector field, and is globally attracting. Within the van der Pol plane, red curve segments depict highly impulsive regions along the attracting limit cycle while blue curve segments depict the more slowly evolving regions. B. Representative trajectory time series for the distance estimator $f(z)$ (blue), the position variable of the anchoring system $(\dot{x}_3, \dot{x}_4)$ (black dashed), and the acceleration of the van der Pol system $\dot{x}_2$ (red dashed). C. Visualizations of the trajectory datasets plotted with respect to their leading three global principal components (GPCs), and colored by magnitude of the distance estimator. Low values of the distance estimator indicate predict the locus of a locally hyperbolic attracting set. The highly impulsive episodes of the van der Pol oscillator disturb the predictions, which, otherwise, accurately locate the limit cycle locus, suggesting the distance estimator's utility for detecting hybrid behavior.
• Figure 5: Illustrative VERT predictions of running control modules are supported by comparison to empirically validated template models based on data withheld from the VERT pipeline. Blue traces show the VERT distance estimate. A. The postural state data input to VERT are presented as trajectories through the space of spatial pairwise distances (and their velocities) between body markers, $z\in \mathbb{R}^{182}$. B. Visualization of the stance phase postural trajectories colored by value of the distance estimator, projected onto the first 3 global principal components (GPCs) of the postural state space. Arrow indicates direction of flow, from impact to lift-off. C. Stance phase averaged time series (with shadowed data traces) of the distance estimator $f(z)$ (blue trace), aligned with vertical ground reaction force (GRF), $F_z$ (red trace, withheld from VERT). D-E. We use VERT to search for control modules in the Arm-Trunk (D) and Leg (E) postural state based only upon projections of the purely kinematic data (visualized in the lower plot of A). As a validation experiment probing VERT's ability to find postures carrying previously identified template dynamics (E), and to suggest novel postures that might carry new templates (D), we align and compare the averaged distance estimator $f(z)$ to the average vertical GRF time series ($F_z$) and kinematics informed by the previously known template hypotheses - the trunk and arm pitch angles ($\psi_t$ and $\psi_a$, respectively, in row D) and the virtual leg length from the COM to heel ($l_v$ in row D). The template phase of stance predicted by VERT's proposal of a stabilized posture submanifold (the blue shaded interval that marks the kinematic data at which $f(z)$ vanishes) aligns with the canonical dynamic behavior of the SLIP model full1999templatesholmes2006dynamics.

Theorems & Definitions (4)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1